The terahertz (THz) beam located from 0.1 to 10 THz opens a new route for signal transmission, light–matter interaction, spectral analysis, imaging, and detection^[1–5]. THz beams can penetrate through opaque non-polar materials and be endowed with a molecular fingerprint, bringing significant applications for safety inspection, nondestructive defect detection, biomedicine trace sensing, and so on^[6–8]. Due to the relatively large wavelength, THz lenses are indispensable for enhancing the spatial resolution in the above applications. The advance of metalenses brings a compact yet flexible platform for THz beam focusing and wavefront shaping^[9,10]. By accurately defining the phase modulation profile using the subwavelength elements, metalenses can be made free of spherical aberration^[11], can be achromatic over a certain bandwidth^[12,13], can have a wide angle, and can provide tight focusing with a near-unity numerical aperture (NA)^[14–16], to name a few advantages.

As compared to the above functions, shape engineering of the focal spot is less studied in metalenses. The shape of the focal spot is characterized by the full width at half-maximum (FWHM) and the depth of focus (DOF). They are both locked to the NA as FWHM=λ/(2NA) and DOF=2nλ/NA2^[17]. The FWHM determines the lateral resolution, while the DOF determines the imaging distance^[16,18]. Long DOF provides a high tolerance of axial positioning. Needle beams with relatively long DOF are generated in metasurfaces based on the azimuthally polarized beam, where extra components are needed to generate the structured illumination^[19,20]. Inverse design based on the optimization algorithm is another way to extend the DOF, which is effective but without a physical explanation^[21,22]. There are few theories or methods to guide the design with a user-defined DOF.

Recently, metasurfaces have been used for the on-demand design of the Jones matrix using meta-molecules composed of two or more meta-atoms^[23–25]. We generalize the application scope of this method from a lossless to a lossy system through matrix decomposition^[26]. These metasurfaces have been demonstrated for polarization-multiplexed holographic imaging^[27], polarimetry^[28], and optical encryption^[29] in multiple channels.

Here, we extend this method to tune the DOF of a metalens in a polarization-multiplexed manner. The metalens is designed to have varied focal lengths in four linear polarization channels, which is done by incorporating four rods in a meta-molecule. The DOF with circularly polarized illumination is enhanced due to the combined contribution of all the linear polarization channels. The absolute focusing intensity and the relative phase in each channel can be accurately tuned through the Jones matrix to enable long and uniform needle beam generation. As a result, the design here involves two interference processes: The Jones matrix is obtained through the coherent interference of four neighboring rods, and the focus on the circularly polarized excitation is the coherent interference of four linear polarization channels. It offers a clear physical mechanism for metalens construction with the desired DOF.

The basic meta-atom is the rectangular rod made of Al2O3 in Fig. 1(a), which is a 3D-printable high-index dielectric material. The refractive index is 3+0.02i at the working frequency of 0.14 THz. The period P and height h are fixed as 1 and 2 mm, respectively. The substrate thickness is 1.5 mm. Without rotation, the Jones matrix of the single rod can be described by Ja=[Jxx,0;0,Jyy]=[Axxexp(iφxx),0;0,Ayyexp(iφyy)]. By tuning the width W and the length L, Axx(Ayy) and φxx(φyy) are schematically shown in Fig. 1(b). To decouple the amplitude and the phase, two rods are combined as Jb=Ja1+Ja2 through the interference. In each diagonal term of Jb, the amplitude can be tuned from 0 to 1 and the phase from 0 to 2π independently. So any diagonal Jones matric can be realized using a rod pair in Fig. 1(a). Our previous study shows that an arbitrary symmetric Jones matrix can be decomposed as two diagonal matrices following^[26]Jc=[JxxJyxJxyJyy]=[Jxx00Jyy]+R[Jyx00−Jxy]R−1.(1)Here R=[cosπ/4,−sinπ/4;sinπ/4,cosπ/4] is the rotation matrix to rotate the diagonal matrix by π/4. The off-diagonal components of its Jones matrix are identical (Jxy=Jyx). As a result, a symmetric Jones matrix can be constructed by combining two sets of rod pairs, with one pair oriented towards 0° and the other towards 45°, as shown in Fig. 1(a).

We randomly generate 100 symmetric Jones matrices as the targets, with the constraints of |Jxx|2+|Jxy|2≤1, |Jyy|2+|Jyx|2≤1. Then we construct them utilizing the alumina supercells composed of four rods. The Euclidean distance between the target Jones parameter and the constructed one is defined as the error summarized in Fig. 1(c). The overall error is in the order of 0.012.

Based on accurate Jones matrix construction, we move to design the metalens for varied focal lengths in different polarization channels. Here, we choose four linear polarization channels, which are x–x, x–y, y–x, and y–y with the first as the input polarization and the second as the output. The response in each channel is determined by one specific Jones parameter distribution, which is the spatial profile of Jxx, Jxy, Jyx, and Jyy, respectively. Due to the symmetry constraints of the Jones matrix in the single-layer metalens, the response in the x–y and y–x channels are set to be the same. The profiles of Jxx, Jxy(Jyx), and Jyy are calculated according to the target focal length of fxx=5.5cm, fxy(fyx)=8cm, and fyy=12cm, respectively, following φuv=2πλ(fuv−X2+Y2+fuv2),(2)where u and v represent the incident and output linear polarization states, respectively. Such a choice of the focal length is to separate the focus as much as possible and still to ensure the end-to-end connection in different channels. Figure 2(a) shows the schematic of the metalens. It is composed of 25×25 meta-molecules and 50×50 rods. The diameter is 5 cm. Lumerical FDTD Solutions is used to calculate the transmission field distributions. A Gaussian beam with a working frequency of 0.14 THz is illuminated from the substrate side. The waist radius is 28 mm. A perfectly matched layer (PML) is placed in the x-, y-, and z-directions. A plane monitor is used to collect the field distribution in the x–z plane. The incident light polarization is given by setting the initial polarization direction angle of the light source, and the output polarization is obtained by extracting the desired polarization components in each channel. The simulated beam pattern in each polarization channel is shown in Figs. 2(b)–2(d), respectively. The focal length in each channel agrees well with the design. Since each channel is determined by one Jones parameter, the focus in each channel does not have crosstalk.

In Figs. 2(b)–2(d), the peak intensity shows large variation among different channels. For clarity, the intensity along the optical axis is plotted in Fig. 3(a) for each channel, called the sub-focus. Since Ixy and Iyx are always identical, we plot the sum of them for clarity. The shorter focal length gives a larger peak intensity. When the same metalens is illuminated by a circularly polarized Gaussian beam of the same size, all four channels are activated, and the output focal spot shows (solid line) apparent non-uniformity. The DOF is not improved as compared to a sub focus.

Next, we will tune the intensity and phase of the sub-focus with a target of generating a uniform and long needle beam with circularly polarized excitation. We first give a weight factor to each linear channel to tune the peak intensity. A shorter focal length results in a larger peak intensity and is, therefore, given a smaller weight for channel balance. After some attempts, we multiply the Jones parameters Jxx, Jxy(Jyx), and Jyy by 0.3, 0.17, and 0.6, respectively. After the intensity modulation, the axial intensity profile in each channel is modified in Fig. 3(b) with similar peak values. But still, the distribution with circularly polarized excitation is not a good needle shape. This is because the right-handed circular polarization (RCP) adds a π/2 phase delay to the y component relative to x. So, the Jones parameters should be modified accordingly by adding the π/2 and −π/2 phase delays to Jxx and Jyy. After these modifications, the distribution of the Jones parameters is shown in Fig. 4. The amplitude is shown by the height and the phase by the color. The metalens is redesigned according to Fig. 4. By doing the full-wave simulation, the electric field intensity profiles in the x–z plane are shown in Figs. 5(a)–5(c) in different channels. When the metalens is illuminated by the RCP beam, the total field intensity distribution is shown in Fig. 5(d). It clearly illustrates the focusing depth ranging from 6 to 14.9 cm. It is defined as half the maximum focused spot energy along the optical axis. The simulated DOF is 41λ.

The metalens was fabricated by Digital Light Processing (DLP) 3D printing technology. First, the photosensitive resins HDDA, HEA, and TMPTA in the 2:1:1 volume ratio were mixed with the photoinitiator TPO thoroughly to obtain an organic solvent. Then, the ceramic powder was dissolved in the organic solvent to obtain the required ceramic slurry. Next, the ceramic slurry was injected into the tray of the printer. A 405 nm ultraviolet light was irradiated on the platform to solidify the ceramic slurry layer by layer until the printing was completed. The slice thickness of each layer was 20 µm, and the exposure time for each layer was 10 s. The laser exposure power density is 7mW/cm2. Finally, the printed ceramic embryo was heated in a degreasing furnace at 600°C for 30 h and in a high-temperature sintering furnace at 1600°C for 20 h to get the ceramic sample. The photograph of the fabricated metalens is shown in Fig. 6(a), with a zoomed-in view in Fig. 6(b). The rods in the yellow dashed square are a composite meta-molecule.

In the experiment, the beam from a continuous-wave IMPATT diode source was collimated by a HDPE lens into a quasi-Gaussian beam. The incident laser is a y-polarization beam. Then, a half-wave plate in front of the metasurface was used to control the incident beam with x- or y-polarization, and a quarter-wave plate was used to control the beam with circular polarization. A linear polarizer was used to filter transmitted polarization for linearly polarization channels and removed for circularly polarized excitation. The intensity profile detection was measured by scanning the detector with a pinhole.

The measured field distributions are shown in Figs. 5(e)–5(h). The measured focal lengths are 5.6, 8.4, and 12.6 cm in the x–x, x–y(y–x), and y–y linear channels. With RCP excitation, the field distribution is a constructive addition of the four linear channels, leading the focus from 6.7 to 16.5 cm. The simulation and experiment are consistent. The axial intensities in each linear polarization channel and in the RCP case are summarized in Figs. 3(c) and 3(d). The FWHM of the needle beam is 5.5 mm, which is close to the diffraction limit of 5.1 mm and constrained by the NA. The focal spot shows 46λ DOF with uniform intensity distribution in the test, validating the effectiveness of this method in tailoring the DOF. The measured DOF is slightly larger than the simulated one, which is attributed to the small deviation of the tested focal length in the linear polarization channels.

In summary, we have proposed a polarization-multiplexed manner to design the metalens with the desired long DOF. The metalens has varied focal lengths in x–x, x–y(y–x), y–y channels. By accurately tuning the focusing intensity and relative phase in each channel, the focal spot with the circularly polarized excitation is the coherent interference in four linear polarization channels. A focal spot with an ultralong focusing distance of 46λ is experimentally demonstrated under the illumination of right-handed circularly polarized light. Using this method, on-demand tailoring of the DOF is possible for various applications.